Our task is to find all central extensions of an algebra \(\bf A\)  by
a space \({\mathbb {V}}\) . In order to solve the isomorphism problem we need to study the
action of \(\operatorname{Aut}({\bf A})\)  on \({\rm H^{2}}\left( {\bf A},{\mathbb {V}}\right) \) . To do that, let us fix a basis \(\lbrace e_{1},\ldots ,e_{s}\rbrace \)  of \({\mathbb {V}}\) , and \(\theta \in {\rm Z^{2}}\left( {\bf A},{\mathbb {V}}\right) \) . Then \(\theta \)  can be uniquely
written as \(\theta \left( x,y\right) =\displaystyle \sum _{i=1}^{s} \theta _{i}\left( x,y\right) e_{i}\) , where \(\theta _{i}\in {\rm Z^{2}}\left( {\bf A},\mathbb {C}\right) \) . Moreover, \(\operatorname{Ann}(\theta )=\operatorname{Ann}(\theta _{1})\cap \operatorname{Ann}(\theta _{2})\cap \ldots \cap \operatorname{Ann}(\theta _{s})\) . Furthermore, \(\theta \in {\rm B^{2}}\left( {\bf A},{\mathbb {V}}\right) \)  if and only if all \(\theta _{i}\in {\rm B^{2}}\left( {\bf A},\mathbb {C}\right) \) .
It is not difficult to prove (see [1]}) that given a Zinbiel algebra \({\bf A}_{\theta }\) , if we write as
above \(\theta \left( x,y\right) = \displaystyle \sum _{i=1}^{s} \theta _{i}\left( x,y\right) e_{i}\in {\rm Z^{2}}\left( {\bf A},{\mathbb {V}}\right) \)  and
\(\operatorname{Ann}(\theta )\cap \operatorname{Ann}\left( {\bf A}\right) =0\) , then \({\bf A}_{\theta }\)  has an
annihilator component if and only if \(\left[ \theta _{1}\right] ,\left[\theta _{2}\right] ,\ldots ,\left[ \theta _{s}\right] \)  are linearly
dependent in \({\rm H^{2}}\left( {\bf A},\mathbb {C}\right) \) .
